Mathematics Question from JEE Main 2025 · 4 April, Shift 2

Let $f(x) + 2f\left( \frac{1}{x} \right) = x^2 + 5$ and $2g(x) - 3g\left( \frac{1}{2} \right) = x, \; x > 0. \; \text{If} \; \alpha = \int_{1}^{2} f(x) \, dx, \; \beta = \int_{1}^{2} g(x) \, dx, \text{ then the value of } 9\alpha + \beta \text{ is:}$